On local property of \(| \overline{N},p_ n;\delta|_ k\) summability of factored Fourier series
Let \(\sum^ \infty_{n=1} A_ n(x)\) denote the Fourier series of a \(2\pi\)-periodic and Lebesgue integrable function. Suppose \(X_ n= P_ n(np_ n)^{-1}\), where \((p_ n)\) is a sequence of positive numbers such that \(P_ n= p_ 0+ p_ 1+\cdots+ p_ n\to\infty\), as \(n\to\infty\). Also write \(\Delta t_ n\) for \(t_ n- t_{n+1}\) for any sequence \((t_ n)\). Recently, the present author [J. Math. Anal. Appl. 163, No. 1, 220-226 (1992; Zbl 0764.42002)] proved that \(|\overline N,p_ n|_ k\) \((k\geq 1)\) is a local property of the series \(\sum^ \infty_{n=1} A_ n(x) X_ n\lambda_ n\), where \(\Delta X_ n=O(1/n)\), \(\sum^ \infty_{n=1} (1+ X^ k_ n)|\Delta \lambda_ n|< \infty\) and \(\sum^ \infty_{n=1} X^{k-1}_ n n^{-1}(|\lambda_ n|^ k+ |\lambda_{n+1}|^ k)< \infty\). In the present paper, the author has extended the above result for the summability \(|\overline N,p_ n;\delta|_ k\) \((k\geq 1,\;\delta\geq 0)\) which reduces to \(|\overline N,p_ n|_ k\) \((k\geq 1)\) for \(\delta= 0\).
scientific article; zbMATH DE number 566476